It follows from Proposition  that when \(V\)  is a finite
dimensional vector space then for some \(n \in \mathbb {N}\) ,
\(\mathrm {ker}(!_{n+1})=\mathrm {ker}(!_n)\)  i.e., the procedure terminates after a
finite number of steps. This is because for vector spaces and linear maps,
kernels (in the sense of [1]}) correspond to subspaces
and for a finite dimensional space there can only be a finite chain of subspace
containment; see [1]} for the details. For the more
general case of finitely generated Artinian semimodules over a semiring, we can
also guarantee stabilization of the cochain. This is shown in the following
proposition.
